#
March 2002
Jlab-Thy-02-07
Suny-Ntg-02-4
Su-4252-755

Vector Meson Dominance Model for

Radiative Decays Involving Light Scalar Mesons

###### Abstract

We study a vector dominance model which predicts a fairly large number of currently interesting decay amplitudes of the types , and , where and denote scalar and vector mesons, in terms of three parameters. As an application, the model makes it easy to study in detail a recent proposal to boost the ratio by including the isospin violating - mixing. However we find that this effect is actually small in our model.

There is increasing interest in a possible nonet of light scalar mesons (all of mass ). In addition to the well established and evidence of both experimental and theoretical nature for a very broad () and a very broad () has been presented kyotoconf . The latter two resonances are difficult to identify cleanly because they appear to be of non Breit-Wigner type, signaling strong interference with the non-resonant background.

Such a nonet would most likely represent meson states more complicated than quark-anti quark type and hence would be of great importance for a full understanding of QCD in its non-perturbative low energy regime.

Clearly it is important to study the properties of the and from the point of view of how they fit into a putative nonet family. In particular, the reactions and have recently been observed recentexpts with good accuracy and are considered as useful probes of scalar properties. The theoretical analysis was initiated by Achasov and Ivanchenko Achasov-Ivanchenko and followed up by many others radiative . The models employed are essentially variants of the single meson loop diagram to which a -type vector meson, a photon and two pseudoscalars or a scalar are attached.

In the present note we introduce a complementary approach which emphasizes the “family” or symmetry aspects of the analysis. This enables us to study the correlations among a fairly large number of related radiative amplitudes in terms of a few parameters, without making a commitment to a particular quark structure for the scalars.

Our framework is that of a standard non-linear chiral Lagrangian containing, in addition to the pseudoscalar nonet matrix field , the vector meson nonet matrix and a scalar nonet matrix field denoted . Under chiral unitary transformations of the three light quarks; , the chiral matrix , where , transforms as . The convenient matrix CCWZ is defined by the following transformation property of (): , and specifies the transformations of “constituent-type” objects. The fields we need transform as

(1) |

where the coupling constant is about . One may refer to Ref. Harada-Schechter for our treatment of the pseudoscalar-vector Lagrangian and to Ref. Black-Fariborz-Sannino-Schechter:99 for the scalar addition. The entire Lagrangian is chiral invariant (modulo the quark mass term induced symmetry breaking pieces) and, when electromagnetism is added, gauge invariant.

It should be remarked that the effect of adding vectors to the chiral Lagrangian of pseudoscalars only is to replace the photon coupling to the charged pseudoscalars as,

(2) |

where is the photon field, and with . The ellipses stand for symmetry breaking corrections. We see that in this model, Sakurai’s vector meson dominance Sakurai simply amounts to the statement that (the KSRF relation KSRF ). This is a reasonable numerical approximation which is essentially stable to the addition of symmetry breakers Harada-Schechter ; foot:VD and we employ it here by neglecting the last term in Eq. (2). Although vector meson dominance must be somewhat modified in cases where the axial anomaly plays a role BKY:88 , it generally works quite well for processes such as those we study here.

The new feature of the present work is the inclusion of strong trilinear scalar-vector-vector terms in the effective Lagrangian:

(3) |

Chiral invariance is evident from (1) and the four flavor-invariants are needed for generality. (A term is linearly dependent on the four shown). Actually the term will not contribute in our model so there are only three relevant parameters , and . Equation (3) is analogous to the interaction which was originally introduced as a coupling a long time ago GSW . It is intended to be a leading point-like nna description of the production mechanism. With (2) one can now compute the amplitudes for and according to the diagrams of Fig. 1.

The decay matrix element for is written as where stands for the photon polarization vector. It is related to the width by

(4) |

and takes on the specific forms:

(5) |

Here , and where the scalar mixing angle, is defined from

(6) |

Furthermore ideal mixing for the vectors, with , , , was assumed for simplicity.

Similarly, the decay matrix element for is written as . It is related to the width by

(7) |

where is the photon momentum in the rest frame. For the energetically allowed processes we have

(8) |

In addition, the same model predicts amplitudes for the energetically allowed processes: , , , and, if is sufficiently heavy . The corresponding width is

(9) |

where and

(10) |

All the different decay amplitudes are described by the parameters , and . The reason does not appear at all and does not appear for is that, noting Eq. (2), the factor is seen to give zero when coupled to an external photon line. Because the and are so broad, the simple two body final state approximation in decays like , is not accurate. It is better to consider these decays as having three body final states with the terms in Eq. (3) giving the vertices and to take into account large width corrections in the scalar propagators as well as non resonant background.

These formulas can be used for different choices of the quark structure of the scalar nonet (e.g. the usual scenario or the “dual” scenario where ). The characteristic mixing angle is expected to differ, depending on the scheme. In the literature, besides conventional models, models Jaffe , meson-meson “molecule” models IW and unitarized meson-meson umm models have been investigated. Recently models featuring mixing between a nonet and a heavier nonet have been proposed mixing ; in this case two sets of interactions like Eq. (3) should be included.

Now we shall illustrate the procedure for the model of a single putative scalar nonet Black-Fariborz-Sannino-Schechter:99 with a mixing angle, (characteristic of type scalars).

The parameters and may be estimated from the processes. Substituting (obtained using PDG ) into Eqs. (4) and (5) yields (assumed positive in sign). Of course, this value is independent of the value of . Then, yields either or . In turn we formally predict to be either or respectively.

Next consider the radiative decays. Assuming is dominated by , and Eq. (8) determines as either or . Note that and are almost an order of magnitude smaller than . Thus, the radiative decay rates are mainly determined by . Knowing , and we can predict using Eq. (8). There are four possibilities due to the two possibilities each for and . The largest number, corresponds explainphasespace to the choice and .

Unfortunately this is still considerably smaller than the listed value PDG : foot:phif0gam . Recently Close and Kirk Close-Kirk proposed that the ratio could be boosted by considering the effects of the isospin violating - mixing. We will now see that these effects are small in our model. One may simply introduce the mixing by a term in the effective Lagrangian: . A recent calculation ABFS for the purpose of finding the effect of the scalar mesons in the process obtained the value . It is convenient to treat this term as a perturbation. Then the amplitude for includes a correction term consisting of the amplitude given in Eq. (8) multiplied by and by the propagator. The amplitude has a similar correction. In terms of the amplitudes in Eq.(8) the desired ratio is then,

(11) |

where and . In this approach the propagators are diagonal in the isospin basis. The numerical values of these resonance widths and masses are, according to the Review of Particle Physics PDG , – MeV, and – MeV. For definiteness, from column 1 of Table II in Ref. Harada-Sannino-Schechter we take and while in Eq. (4.2) of Ref. FS we take . In fact the main conclusion does not depend on these precise values. It is easy to see that the mixing factors are approximately given by

(12) |

Noting that in the present model, the ratio in Eq.(11) is roughly . Clearly, the correction to due to - mixing only amounts to a few per cent, nowhere near the huge effect suggested in Close-Kirk . It may be remarked that Eq.(11) is practically accurate to all orders in , corresponding to iterating any number of - transitions. Then, after summing a geometric series, the numerator picks up a correction factor and the denominator, the similar factor .

Vector meson dominance, together with the assumptions of flavor symmetry and a single nonet of scalar mesons makes many more predictions. These are listed in Table I for two of the allowed parameter sets, neglecting - mixing. It will be interesting to see if future experiments confirm the pattern of predicted widths.

ratio | ||
---|---|---|

We have given a leading order correlation of many radiative decays involving scalars, based on flavor symmetry and vector meson dominance. Clearly further improvements can be made. Elsewhere, we will study flavor symmetry breaking effects, higher drivative interaction terms, treatment of the final states as , and the case of mixed and scalar nonets.

We are happy to thank N. N. Achasov for a suggested correction to the first version of this note and with A. Abdel-Rahiem and A. H. Fariborz for very helpful discussions. D.B. wishes to acknowledge support from the Thomas Jefferson National Accelerator Facility operated by the Southeastern Universities Research Association (SURA) under DOE contract number DE-AC05-84ER40150. The work of M.H. is supported in part by Grant-in-Aid for Scientific Research (A)#12740144 and USDOE Grant Number DE-FG02-88ER40388. The work of J.S. is supported in part by DOE contract DE-FG-02-85ER40231.

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